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Let `nge3`. A list of numbers `x_(1),x_(2),....,x_(n)` has mean `mu` and standard deviation `sigma` . A new list of numbers `y_(1),(y_(2),....,y_(n)` is made as follows `: y_(1)=(x_(1)+ x_(2))/(2),y_(2)=(x_(1)+x_(2))/(2) and y_(j)"for"j=3,4,....,n`. The mean and the standard deviation of the new list are `hatmuand hatsigma` . Then whcih of the following is necessarily true?A. `mu=hatmu and sigmalehat sigma`B. `mu=hatmu and sigma ge hat sigma`C. `sigma-hat sigma`D. `munehat mu` |
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Answer» Correct Answer - B `mu=(x _(1)+x_(2)+.....x_(n))/(n)` `hatmu=(y_(1)+y_(2)+....+y_(n))/(n)=((x_(1)+x_(2))/(2)+(x_(1)+x_(2))/(2)+x_(3)+....+x_(n))/(n)` `hatmu=(x_(1)+x_(2)+....+x_(n))/(n)=murArrhatmu=mu` `sigma^(2)=(sumx_(i)^(2))/(n)-mu^(2)` `sigma^(2)=(x_(1)^(2)+x_(2)^(2)+....+x_(n)^(2))/(n)-mu^(2)" "....(1)` `hatsigma^(2)=(x_(1)^(2)+y_(2)^(2)+....+y_(n)^(2))/(n)-mu^(2) (((x_(1)+x_(2))/(2))^(2)+((x_(1)+x_(2))/(2))^(2)+x_(3)^(2)+....x+_(n)^(2))/(n)-mu^(2)` `hatsigma^(2)=((x_(1)^(2)+x_(2)^(2))/(2)+x_(1)x_(2)+x_(3)^(2)+....+x_(n)^(2))/(n)-mu^(2)" " ....(2)` `sigma^(2)-hatsigma^(2)=(x_(1)^(2)+x_(2)^(2))/(n)-((x_(1)^(2)+x_(2)^(2)+2x_(1)x_(2))/(2n))=(x_(1)^(2)+x_(2)^(3)-2x_(1)x_(2))/(2n)` `(x _(1)-x_(2))^(2)/(2n)ge0rArrgehat sigma&mu=hatmu` |
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